B-spline representation of optical surfaces and its accuracy in a ray trace algorithm

Philipp Jester; Christoph Menke; Karsten Urban

doi:10.1364/AO.50.000822

Applied Optics
Vol. 50,
Issue 6,
pp. 822-828
(2011)
•https://doi.org/10.1364/AO.50.000822

B-spline representation of optical surfaces and its accuracy in a ray trace algorithm

Philipp Jester, Christoph Menke, and Karsten Urban

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Philipp Jester, Christoph Menke, and Karsten Urban, "B-spline representation of optical surfaces and its accuracy in a ray trace algorithm," Appl. Opt. 50, 822-828 (2011)

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Abstract

We introduce a representation of aspheric surfaces that is based on a B-spline quasi-interpolation scheme. The scheme is implemented in a ray trace algorithm, and bounds on the approximation error are established. Examples for the reproduction of aspheric surfaces in polynomial description and the ray tracing accuracy are presented. The proposed approach allows the specification of local and global structures and the efficient treatment of measured surface data. The representation gives access to a wavelet analysis, offering extended possibilities for the tolerance analysis of optical systems containing aspheric elements.

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KXY		${KSA}_{1}$		${KSA}_{2}$
b	$5.0 mm$	b	$1.0 mm$	b	$20.0 mm$
$ρ_{x}$	$- 2.15 e^{- 1}$	ρ	1.00	ρ	$- 3.87 e^{- 2}$
$ρ_{y}$	$1.22 e^{- 1}$	—	—	—	—
$κ_{x}$	$- 1.00$	κ	$- 1.00$	κ	0.00
$κ_{y}$	$- 1.00$	—	—	—	—
$c_{10}$	$- 4.05 e^{- 4}$	$c_{1}$	1.50	$c_{1}$	$- 4.17 e^{- 6}$
$c_{12}$	$- 8.13 e^{- 4}$	$c_{2}$	$- 7.00 e^{- 1}$	$c_{2}$	$4.71 e^{- 9}$
$c_{14}$	$5.73 e^{- 4}$	$c_{3}$	$5.00 e^{- 1}$	$c_{3}$	$- 4.94 e^{- 12}$
$c_{21}$	$- 4.59 e^{- 6}$	$c_{4}$	$- 5.00 e^{- 1}$	$c_{4}$	$- 5.42 e^{- 15}$
$c_{23}$	$1.14 e^{- 5}$	—	—	$c_{5}$	$- 4.98 e^{- 18}$
$c_{25}$	$9.64 e^{- 6}$	—	—	$c_{6}$	$- 1.22 e^{- 20}$
$c_{27}$	$4.45 e^{- 7}$	—	—	—	—
$c_{36}$	$- 2.69 e^{- 9}$	—	—	—	—
$c_{38}$	$- 7.96 e^{- 8}$	—	—	—	—
$c_{40}$	$- 8.79 e^{- 8}$	—	—	—	—
$c_{42}$	$- 9.16 e^{- 8}$	—	—	—	—
$c_{44}$	$2.43 e^{- 8}$	—	—	—	—

d	$j = 8$	$j = 9$	$j = 10$
1	$3 s$	$3 s$	$4 s$
2	$3 s$	$4 s$	$6 s$
3	$5 s$	$6 s$	$7 s$
4	$8 s$	$9 s$	$11 s$
5	$12 s$	$14 s$	$15 s$
6	$18 s$	$20 s$	$23 s$

KXY		${KSA}_{1}$		${KSA}_{2}$
b	$5.0 mm$	b	$1.0 mm$	b	$20.0 mm$
$ρ_{x}$	$- 2.15 e^{- 1}$	ρ	1.00	ρ	$- 3.87 e^{- 2}$
$ρ_{y}$	$1.22 e^{- 1}$	—	—	—	—
$κ_{x}$	$- 1.00$	κ	$- 1.00$	κ	0.00
$κ_{y}$	$- 1.00$	—	—	—	—
$c_{10}$	$- 4.05 e^{- 4}$	$c_{1}$	1.50	$c_{1}$	$- 4.17 e^{- 6}$
$c_{12}$	$- 8.13 e^{- 4}$	$c_{2}$	$- 7.00 e^{- 1}$	$c_{2}$	$4.71 e^{- 9}$
$c_{14}$	$5.73 e^{- 4}$	$c_{3}$	$5.00 e^{- 1}$	$c_{3}$	$- 4.94 e^{- 12}$
$c_{21}$	$- 4.59 e^{- 6}$	$c_{4}$	$- 5.00 e^{- 1}$	$c_{4}$	$- 5.42 e^{- 15}$
$c_{23}$	$1.14 e^{- 5}$	—	—	$c_{5}$	$- 4.98 e^{- 18}$
$c_{25}$	$9.64 e^{- 6}$	—	—	$c_{6}$	$- 1.22 e^{- 20}$
$c_{27}$	$4.45 e^{- 7}$	—	—	—	—
$c_{36}$	$- 2.69 e^{- 9}$	—	—	—	—
$c_{38}$	$- 7.96 e^{- 8}$	—	—	—	—
$c_{40}$	$- 8.79 e^{- 8}$	—	—	—	—
$c_{42}$	$- 9.16 e^{- 8}$	—	—	—	—
$c_{44}$	$2.43 e^{- 8}$	—	—	—	—

d	$j = 8$	$j = 9$	$j = 10$
1	$3 s$	$3 s$	$4 s$
2	$3 s$	$4 s$	$6 s$
3	$5 s$	$6 s$	$7 s$
4	$8 s$	$9 s$	$11 s$
5	$12 s$	$14 s$	$15 s$
6	$18 s$	$20 s$	$23 s$

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Equations (24)

Applied Optics